Question

В круге проведены диаметр AB и параллельные хорды AC и BT. Докажите ,что CB=TA. Докажите ,что CB=TA

Answer 1

Треугольники АСВ и АТВ прямоугольные, угол Т=углуС=90 - вписанные углы, опирающиеся на диаметр = 180/2=90, треугольники равны по гипотенузу АВ - общая и острому углу, угол АВТ=углу ВАС как внутренние разносторонние. СВ=ТА

Answer 2

Given Information:

In the given figure, a circle is drawn with diameter AB. Two parallel chords AC and BT are also drawn.

To Prove:

We need to prove that CB = TA.

Proof:

To prove that CB = TA, we can use the property of parallel chords in a circle. According to this property, if two chords in a circle are parallel, then they intercept congruent arcs.

In this case, chords AC and BT are parallel, so they intercept congruent arcs. Let's denote the arcs intercepted by these chords as arc AC and arc BT.

Now, since AB is the diameter of the circle, it divides the circle into two semicircles. Let's denote the semicircles as semicircle AC and semicircle BT.

Since arc AC and arc BT are congruent, it means that they are arcs of equal length in their respective semicircles.

Now, let's consider triangle ABC. Since AB is the diameter of the circle, it is also the hypotenuse of triangle ABC. By the Pythagorean theorem, we know that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

So, we have: AC^2 + CB^2 = AB^2 ---(1)

Similarly, let's consider triangle BAT. Since AB is the diameter of the circle, it is also the hypotenuse of triangle BAT. By the Pythagorean theorem, we have: TA^2 + BT^2 = AB^2 ---(2)

From equations (1) and (2), we can see that AC^2 + CB^2 = TA^2 + BT^2.

Since arc AC and arc BT are congruent, it means that they are arcs of equal length in their respective semicircles. Therefore, the lengths of AC and BT are equal.

So, we can rewrite equation (1) as: AC^2 + CB^2 = TA^2 + CB^2

By canceling out the common term CB^2 from both sides, we get: AC^2 = TA^2

Taking the square root of both sides, we have: AC = TA

Since AC = TA and the lengths of AC and BT are equal, it implies that CB = TA.

Hence, we have proved that CB = TA.